Osaka Journal of Mathematics

Active sums II

Alejandro J. Díaz-Barriga, Francisco González-Acuña, Francisco Marmolejo, and Leopoldo Román

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We exhibit several finite groups that are not active sums of cyclic subgroups. We show that this is the case for groups with $H_{1}G$ of odd order and $H_{2}G$ of even order. As particular examples of this we have the alternating groups $A_n$ for $n\geq 4$, some special and some projective linear groups. Our next set of examples consists of $p$-groups where the normalizer and the centralizer of every element coincide. We also have an example of a 2-group where the above conditions are not satisfied; thus we had to devise an ad hoc argument. We observe that the examples of $p$-groups given also provide groups that are not molecular.

Article information

Osaka J. Math., Volume 43, Number 2 (2006), 371-399.

First available in Project Euclid: 6 July 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J05: Homological methods in group theory 20D99: None of the above, but in this section
Secondary: 20D30: Series and lattices of subgroups


Díaz-Barriga, Alejandro J.; González-Acuña, Francisco; Marmolejo, Francisco; Román, Leopoldo. Active sums II. Osaka J. Math. 43 (2006), no. 2, 371--399.

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